Optimal. Leaf size=153 \[ \frac{a^3 (A+6 B) \cos (e+f x)}{c^3 f}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}+\frac{2 a^3 c^3 (A+6 B) \cos ^3(e+f x)}{3 f \left (c^3-c^3 \sin (e+f x)\right )^2}-\frac{a^3 x (A+6 B)}{c^3}-\frac{2 a^3 c (A+6 B) \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4} \]
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Rubi [A] time = 0.34216, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {2967, 2859, 2680, 2682, 8} \[ \frac{a^3 (A+6 B) \cos (e+f x)}{c^3 f}+\frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}+\frac{2 a^3 c^3 (A+6 B) \cos ^3(e+f x)}{3 f \left (c^3-c^3 \sin (e+f x)\right )^2}-\frac{a^3 x (A+6 B)}{c^3}-\frac{2 a^3 c (A+6 B) \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2859
Rule 2680
Rule 2682
Rule 8
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac{1}{5} \left (a^3 (A+6 B) c^2\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac{2 a^3 (A+6 B) c \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4}+\frac{1}{3} \left (a^3 (A+6 B)\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^3} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac{2 a^3 (A+6 B) c \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4}+\frac{2 a^3 (A+6 B) \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^2}-\frac{\left (a^3 (A+6 B)\right ) \int \frac{\cos ^2(e+f x)}{c-c \sin (e+f x)} \, dx}{c^2}\\ &=\frac{a^3 (A+6 B) \cos (e+f x)}{c^3 f}+\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac{2 a^3 (A+6 B) c \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4}+\frac{2 a^3 (A+6 B) \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^2}-\frac{\left (a^3 (A+6 B)\right ) \int 1 \, dx}{c^3}\\ &=-\frac{a^3 (A+6 B) x}{c^3}+\frac{a^3 (A+6 B) \cos (e+f x)}{c^3 f}+\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac{2 a^3 (A+6 B) c \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4}+\frac{2 a^3 (A+6 B) \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^2}\\ \end{align*}
Mathematica [B] time = 1.06856, size = 316, normalized size = 2.07 \[ \frac{a^3 (\sin (e+f x)+1)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (48 (A+B) \sin \left (\frac{1}{2} (e+f x)\right )-15 (A+6 B) (e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5+4 (23 A+93 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4-4 (11 A+21 B) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3-8 (11 A+21 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+24 (A+B) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+15 B \cos (e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5\right )}{15 f (c-c \sin (e+f x))^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.138, size = 323, normalized size = 2.1 \begin{align*} -4\,{\frac{A{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-12\,{\frac{B{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) }}-8\,{\frac{A{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}+8\,{\frac{B{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-{\frac{64\,A{a}^{3}}{5\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-5}}-{\frac{64\,B{a}^{3}}{5\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-5}}-{\frac{80\,A{a}^{3}}{3\,f{c}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-3}}-16\,{\frac{B{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-32\,{\frac{A{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-32\,{\frac{B{a}^{3}}{f{c}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}+2\,{\frac{B{a}^{3}}{f{c}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{{a}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) A}{f{c}^{3}}}-12\,{\frac{{a}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) B}{f{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.637, size = 2275, normalized size = 14.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45899, size = 821, normalized size = 5.37 \begin{align*} \frac{15 \, B a^{3} \cos \left (f x + e\right )^{4} + 60 \,{\left (A + 6 \, B\right )} a^{3} f x - 24 \,{\left (A + B\right )} a^{3} -{\left (15 \,{\left (A + 6 \, B\right )} a^{3} f x -{\left (46 \, A + 231 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{3} -{\left (45 \,{\left (A + 6 \, B\right )} a^{3} f x + 2 \,{\left (A + 66 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (5 \,{\left (A + 6 \, B\right )} a^{3} f x - 2 \,{\left (6 \, A + 31 \, B\right )} a^{3}\right )} \cos \left (f x + e\right ) -{\left (15 \, B a^{3} \cos \left (f x + e\right )^{3} + 60 \,{\left (A + 6 \, B\right )} a^{3} f x + 24 \,{\left (A + B\right )} a^{3} -{\left (15 \,{\left (A + 6 \, B\right )} a^{3} f x + 2 \,{\left (23 \, A + 108 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (5 \,{\left (A + 6 \, B\right )} a^{3} f x - 2 \,{\left (4 \, A + 29 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \,{\left (c^{3} f \cos \left (f x + e\right )^{3} + 3 \, c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f -{\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23816, size = 305, normalized size = 1.99 \begin{align*} \frac{\frac{30 \, B a^{3}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )} c^{3}} - \frac{15 \,{\left (A a^{3} + 6 \, B a^{3}\right )}{\left (f x + e\right )}}{c^{3}} - \frac{4 \,{\left (15 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 45 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 30 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 210 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 100 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 420 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 50 \, A a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 270 \, B a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 13 \, A a^{3} + 63 \, B a^{3}\right )}}{c^{3}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{5}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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